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Re: CBSE 2012 Solved Question Paper Class 12
The Central Board of Secondary Education is a Board of Education for public and private schools, under the Union Government of India. The CBSE was formed in 1962. You are looking for the Central Board of Secondary Education class 12th Maths, Physics solved question paper, here i am providing: Physics:Unstable equilibrium when the electric filed is directed at an angle with the dipole axis i.e., when E is not parallel to p . Which part of electromagnetic spectrum is used in radar systems? (1) Solution: The microwave range of electromagnetic spectrum is used in radar systems. Calculate the speed of light in a medium whose critical angle is 30°. A glass lens of refractive index 1.45 disappears when immersed in a liquid. What is the value of refractive index of the liquid? (1) Write the expression for Bohr’s radius in hydrogen atom A wire of resistance 8R is bent in the form of a circle. What is the effective resistance between the ends of a diameter AB? A coil Q is connected to low voltage bulb B and placed near another coil P as shown in the figure. Give reasons to explain the following observations: (2) In a meter bridge, the null point is found at a distance of A cm from A. If now a resistance of X is connected in parallel with S, the null point occurs at /2 cm. Obtain a formula for X in terms of /1, /2 and S. Maths Sample Paper with Solutions Instructions 1. Questions 1 to 10 carry 1 mark each. 2. Questions 11 to 22 carry 4 marks each. 3. Questions 23 to 29 carry 6 marks each. 1. Evaluate: Sol. Let I = Taking x as first function and sin 3x as second function and integrating by parts, we obtain 2. Find the principal value of tan−1 (−1). Sol. Let tan−1 (−1) = y. Then, tan y = −1 = −tan(π/4) = tan(π/4) We know that the range of the principal value branch of tan−1 is Therefore, the principal value of tan−1 (−1) is (-π)/4. 3. Show that * : R × R → R given by a * b → a + 2b is not associative. Sol. The operation * is not associative, since (8 * 5) * 3 = (8 + 10) * 3 = (8 + 10) + 6 = 24, While 8 * (5 * 3) = 8 * (5 + 6) = 8 * 11 = 8 + 22 = 30. 4. If A and B are symmetric matrices of the same order, prove that AB + BA is symmetric. Sol. Let P = AB + BA P′ = (AB + BA)′ = (AB)′ + (BA)′ = B′A′ + A′B′ = BA + AB [ A′=A,B′ = B] = AB + BA = P. 5. Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector Sol. It is given that the line passes through the point A (1, 2, 3). Therefore, the position vector through A is It is known that the line which passes through point A and parallel to is given by is a constant. This is the required equation of the line. 6. Evaluate: Sol. As sin7 (−x) = (sin (−x))7 = (−sin x)7 = −sin7x, therefore, sin7x is an odd function. It is known that, if f(x) is an odd function, then 7. If matrix A = (1,2,3) , write (AA′) , where A′ is the transpose of matrix A. Sol. AA′ = (1 2 3) = (1×1 + 2×2 + 3×3) = (1 + 4 + 9) = 14. 8. Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2). Sol. The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, – 2) is given by, 9. If , then for what value of α is A an identity matrix? Sol. If A is an identity matrix, then: Thus, for α = 0°, A is an identity matrix. 10. The line y = mx + 1 is a tangent to the curve y2 = 4x .Find the value of m. Sol. The equation of the tangent to the given curve is y = mx + 1. Now, substituting y = mx + 1 in y2 = 4x, we get: ⇒ (mx + 1)2 = 4x ⇒ m2x2 + 1 + 2mx – 4x = 0 ⇒ m2x2 + x (2m – 4) + 1 =0 …(i) Since a tangent touches the curve at one point, the roots of equation (i) must be equal. Therefore, we have: Discriminant = 0 (2m – 4) 2 - 4(m2)(1) = ⇒ 4m2 + 16 - 16m – 4m2 = 0 ⇒ 16 – 16m = 0 ⇒ m = 1 Hence, the required value of m is 1. 11. Find : Sol. The given relationship is Differentiating this relationship with respect to x, we obtain 12. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive. Sol. Let A = {1, 2, 3}. A relation R on A is defined as R = {(1, 2), (2, 1)}. It is seen that (1, 1), (2, 2), (3, 3) ∉R. ∴ R is not reflexive. Now, as (1, 2) ∈ R and (2, 1) ∈ R, then R is symmetric. Now, (1, 2) and (2, 1) ∈ R However, (1, 1) ∉ R ∴ R is not transitive. Hence, R is symmetric but neither reflexive nor transitive. 13. Solve the following differential equation: (x2 − y2) dx + 2 xy dy = 0. Sol. Given that y = 1 when x = 1 It is a homogeneous differential equation. Substituting (2) and (3) in (1), we get: Integrating both sides, we get: It is given that when x = 1, y = 1 (1)2+ (1)2 = C (1) ⇒ C = 2 Thus, the required equation is y2 + x2 = 2x. 14. Prove Sol. Let x = sinθ. Then, sin-1x = θ We have, R.H.S. = = sin-1 (sin 3θ) = 3θ = 3 sin-1 x = L.H.S. 15. Evaluate: Sol. The given integral is I = [as sin 2x = 2sin x cos x] On putting (a + b cos x) = t ⇒ −b sin x dx = dt, we obtain where C is a constant 16. Let A be a nonsingular square matrix of order 3 × 3. Then find Sol. We know that, 17. By using properties of determinants, show that: Sol. Taking out common factors a, b, and c from R1, R2, and R3 respectively, we have: Applying R2 → R2 − R1 and R3 → R3 − R1, we have: Applying C1 → aC1, C2 → bC2, and C3 → cC3, we have: Expanding along R3, we have: Hence, the given result is proved. 18. Find the area of the region lying in the first quadrant and bounded by y = 4x2, x = 0, y = 1 and y = 4. Sol. he area in the first quadrant bounded by y = 4x2, x = 0, y = 1, and y = 4 is represented by the shaded area ABCDA as 19. The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7. Sol. Marginal revenue is the rate of change of total revenue with respect to the number of units sold. ∴ Marginal Revenue (MR) = 13(2x) + 26 = 26x + 26 When x = 7, MR = 26(7) + 26 = 182 + 26 = 208 Hence, the required marginal revenue is Rs 208. 20. Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector . Sol. The normal vector is, It is known that the equation of the plane with position vector is given by, This is the vector equation of the required plane. 21. On a multiple choice examination with three possible answers (out of which only one is correct) for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing? Sol. The repeated guessing of correct answers from multiple choice questions are Bernoulli trials. Let X represent the number of correct answers by guessing in the set of multiple choice questions. Let p(a correct answer) = Clearly, X has a binomial distribution with n = 5 and P (guessing more than 4 correct answers) = P(X ≥ 4) = P(X = 4) + P(X = 5) Thus, the required probability is 22. If O be the origin and the coordinates of P be (1, 2, −3), then find the equation of the plane passing through P and perpendicular to OP. Sol. The coordinates of the points, O and P, are (0, 0, 0) and (1, 2, −3) respectively. Therefore, the direction ratios of OP are (1 − 0) = 1, (2 − 0) = 2, and (−3 − 0) = −3 It is known that the equation of the plane passing through the point (x1, y1 z1) is where, a, b, and c are the direction ratios of normal. Here, the direction ratios of normal are 1, 2, and −3 and the point P is (1, 2, −3). Thus, the equation of the required plane is 23. If a young man rides his motorcycle at 25 km/hour, he had to spend Rs 2 per km on petrol. If he rides at a faster speed of 40 km/hour, the petrol cost increases at Rs 5 per km. He has Rs 100 to spend on petrol and wishes to findwhat is the maximum distance he can travel within one hour. Express this as an LPP and solve it graphically. Sol. Let the young man ride with the speed 25 km/hr for x hours and with the speed 40 km/hr for y hours respectively. Money spent at speed 25 km/hr = 50x And Money spent at speed 40 km/hr = 200y Maximum distance : Z = 25x + 40y ….(1) subject to the constraints x + y ≤ 1 ….(2) 50x + 200y ≤ 100 ….(3) x , y ≥ 0 ….(4) Graph the inequalities (2) to (4). The feasible region (shaded) is shown as below: Corner Point Z = 25x + 40y A (0, 1/2) 20 B (2/3, 1/3) 30Maximum C(1, 0) 25 Thus, the maximum distance he can travel is30 km. 24. Evaluate: . Sol. We have Dividing numerator and denominator by x2 25. Solve system of linear equations, using matrix method. x − y + 2z = 7 3x + 4y − 5z = −5 2x − y + 3z = 12 Sol. The given system of equations can be written in the form of AX = B, where Thus, A is non-singular. Therefore, its inverse exists. 26. A family has 2 children. Find the probability that both are boys, if it is known that (i) at least one of the children in a boy, (ii) the elder child is a boy. Sol. Let b stands for boy and g for girl. The sample space of the experiment is (i) Let E and F denote the following events: E: both the children are boys F: at least one of the children is a boy Then, E = {(b, b)} and F = {(b, b), (g, b), (b, g)} (ii) Let P and Q denote the following events: P: both the children are boys Q: elder child is a boy Then, P = {(b, b)} and Q = {(b, b,), (g, b)} 27. Find the equation of the plane passing through the points (1, 2, 3) and (0, −1, 0) and parallel to the line Sol. Equation of any plane passing through the point (1, 2, 3) is given by: a (x − 1) + b (y − 2) + c (z − 3) = 0 … (1) This plane is parallel to the line ∴ a × 2 + b × 3 + c × (−3) = 0 ⇒ 2a + 3b −3c = 0 … (2) Also, plane (1) passes through the point (0, −1, 0). Therefore, a (0 − 1) + b (−1 − 2) + c (0 − 3) = 0 ⇒ −a − 3b − 3c = 0 ⇒ a + 3b + 3c = 0 … (3) Solving equations (2) and (3), we obtain Hence, the required equation of the plane is given by, 6(x − 1) −3 (y − 2) + 1 (z − 3) = 0 ⇒ 6x − 6 −3y + 6 + z − 3 = 0 ⇒ 6x −3y + z − 3 = 0 28. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm. Sol. The volume of a sphere (V) with radius (r) is given by, ∴ Rate of change of volume (V) with respect to time (t) is given by, [By chain rule] It is given that Therefore, when radius = 15 cm, Hence, the rate at which the radius of the balloon increases when the radius is 15 cm is 29. Using integration, find the area of the triangle ABC, coordinates of whose vertices are A (4, 1), B (6, 6) and C (8, 4). Sol. AK, BL, and CM are drawn perpendicular to the x-axis. It can be observed in the following figure that: Area (ΔABC) = Area (AKLB) + Area (BLMC) − Area (AKMC) … (1)
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Re: CBSE Solved Question Paper Class 12
here I am giving you question paper for class 12th examination of Central Board of Secondary Education in PDF file with it so you can get it easily.. class 12th English paper : Section B: Advanced Writing Skills 35 marks Q.3 A.K International School is looking for a receptionist for the school. Write an advertisement on behalf of the adminstrative officer in the classified columns of the local newspaper giving necessary details. Draft the advertisement in not more than 50 words. OR Suman/Suresh has cleared the Pre-Medical Pre-Dental entrance examination. The family is elated at the achievement and they decide to have a get-together for all friends. Draft an informal invitation for the get-together. Q.4. You are Shekhar/Tripta a student of A.P Public School. Principals of two schools from Pakistan visited your school as part of a cultural exchange programme. Students of the school put up a cultural show in their honour. Write a report about it for your school magazine. (100-125 words). OR As you were driving back home from work you were witness to an accident between a Maruti car and a truck. The driver of the car was seriously injured. There was confusion and chaos prevailing on the road. Describe the scene in about 100 to 125 words. You are Sameer/Samiksha. 5. You are Nitin/Natasha a student of Class XII at K.P.N. Public School Faridabad. The student is required to cope with lot of pressure in today’s competitive environment. Write a letter to the editor of a national daily highlighting the increasing stress faced by students and suggest ways to combat the same. OR You are Suresh/Smita. You come across the following advertisement in a national daily. You consider yourself suitable and eligible for the post. Write an application in response to the advertisement. Applications are invited for the post of a N 6. Some colleges conduct entrance test for admission to under-graduate courses like English (Hons.) and Journalism (Hons.). Do you think that the entrance test is the right method of selecting students? Write an article in about 150-200 words. You are Rohan/Rachita, a student of class XII at A.P. International School Agra. OR Computer games and video games have become popular with children today. As a result outdoor games seem to have no place in their life anymore. You are Satish/Sakshi. You had the opportunity of playing Hide-n-Seek when you visited your cousins in a small town. You decide to write an article on your experiences about the joys of playing outdoor games for the school magazine. Write the article in 150-200 words. Q.7. They do not fear the men beneath the tree; They pace in sleek chivalric certainty. a. Are Aunt Jennifer’s tigers real ? Give reasons for your answer. 2 b. Why do the tigers not fear the man beneath the tree? 1 c. What do you understand by ‘chivalric certainty’? 1 Or A thing of beauty is a joy for ever Its loveliness increases, it will never Pass into nothingness; but will keep A bower quiet for us. a. ‘A thing of beauty is joy for ever’. Explain. 2 b. Why does a beautiful thing ‘pass into nothingness’? 1 c. What does poet mean by ‘a bower quiet for us’ 1
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